580 Mb. stokes, on THE CRITICAL VALUES OF 



when n is odd, and B„ = when n is even. If then we omit B^, which is useless, and put for X 

 its value, we get 



— r" -r' 

 _, «x nTrf , 



<t> = 4«)a2o ;; rcos (119)- 



Wtt' - 4a a 



The series multiplied by r^ may be summed. For if we expand sin 2 (0 - ia) between the 

 limits 6 = 0, = a in a series of cosines, we get 



. ^ „ „ 8a cos a tnrO 



sin (2 - a) = - So — cos ; 



H wir0 

 r " cos 



whence (b = Swa^l. -— IT + — - — r''sin(20-a) (120). 



^ W TT («' tt' - 4 a ) 2 cos a 



In determining the motion of the cylinder, the only quantity we care to know is the moment 

 of the fluid pressures about the axis. Now if the motion be so small that we may omit the square 

 of the velocity we shall have, putting (b = — <«/('■) Q), 



dw 

 P = f (0 + -^/{r, 9), 



where p is the pressure, ■i^{t) a function of the time t, whose value is not required, and where 

 the density is supposed to be 1, and the pressure due to gravity is omitted, since it may be taken 

 account of separately. The moment of the pressure on the curved surface is zero, since the 

 direction of the pressure at any point passes through the axis. The expression (ug) or (120) shows 

 that the moments on the plane faces of the sector are equal, and act in the same direction ; so that 

 it will be sufficient to find the moment on one of these faces and double the result. If we consider 

 a portion of the face for which = whose length in the direction of the axis is unity, we shall 



have for the pressure on an element dr of the surface — /(r, 0)dr; and if we denote the whole 



dt 



moment of the pressures by — C — , reckoned positive when it tends to make the cylinder move 



dt 



in the direction of 6 positive, we shall have 



C = 2 /" /(r, 0)rdr. 



•'o 



Taking now the value of /(r, 0) from (120), and performing the integration, we shall have 



C = itana - l6a'2„; -^— (l2l). 



4 °(wir-2a)W7r(w7r + 2a)' 



The mass of the portion of fluid considered is la ; and if we put 



C = lak'\ 



Sir 

 and write — for a, so that s may have any value from to 4, we shall have 



.„ 1 «T 8«^ 1 



k'= — tan 2„ (122). 



«7r 2 ,r^ °(w-s)n(n+«)' ^ "' 



